Abnormal Optimal Trajectory Planning of Multi-Body Systems in the Presence of Holonomic and Nonholonomic Constraints

Abstract

In optimal control problems, the Hamiltonian function is given by the weighted sum of the integrand of the cost function and the dynamic equation. The coefficient multiplying the integrand of the cost function is either zero or one; and if this coefficient is zero, then the optimal control problem is known as abnormal; otherwise it is normal. This paper provides a characterization of the abnormal optimal control problem for multi-body mechanical systems, subject to external forces and moments, and holonomic and nonholonomic constraints. This study does not only account for first-order necessary conditions, such as Pontryagin’s principle, but also for higher-order conditions, which allow the analysis of singular optimal controls.

This is a preview of subscription content, log in to check access.

References

  1. 1.

    Agrachev, A., Sarychev, A.: On abnormal extremals for Lagrange variational problems. J. Math. Syst. Estimation Control 8, 87–118 (1998). doi:10.1145/1731022.1731032

    MathSciNet  MATH  Google Scholar 

  2. 2.

    Aoude, G. S.: Two-Stage Path Planning Approach for Designing Multiple Spacecraft Reconfiguration Maneuvers and Application to SPHERES Onboard ISS. Master’s Thesis, Massachusetts Institute of Technology (2007). MS Thesis, Massachusetts Institute of Technology, Cambridge, MA

  3. 3.

    Arnol’d, V. Encyclopaedia of mathematical sciences: Dynamical systems III. Springer, New York (1988)

  4. 4.

    Barbero-Linán, M., de León, M., de Diego, D. M., Marrero, J. C., Munoz-Lecanda, M. C.: Kinematic reduction and the Hamilton-Jacobi equation. J. Geo. Mech. 4(3), 207–237 (2012). doi:10.3934/jgm.2012.4.207

  5. 5.

    Becerra, V. M. Tech. Rep.: PSOPT Optimal Control Solver User Manual. Reading, United Kingdom (2010)

  6. 6.

    Bell, D. J., Jacobson, D. H.: Singular Optimal Control Problems. Academic Press, New York (1975)

    Google Scholar 

  7. 7.

    Bershanskiy, Y. M.: Conjugation of singular and nonsingular parts of optimal control. Autom. Remote. Control. 40, 325–330 (1979)

    Google Scholar 

  8. 8.

    Bloch, A., Krishnaprasad, P., Marsden, J., Murray, R.: Nonholonomic mechanical systems with symmetry. Arch. Ration. Mech. Anal. 136, 21–99 (1996). doi:10.1007/BF02199365

    MathSciNet  Article  MATH  Google Scholar 

  9. 9.

    Bloch, A. M., Crouch, P. E.: Reduction of Euler Lagrange problems for constrained variational problems and relation with optimal control problems Proceedings of the 33rd IEEE Conference on Decision and Control. doi:10.1109/CDC.1994.411534, vol. 3, pp 2584–2590 (1994)

  10. 10.

    Bloch, A. M., Marsden, J. E., Zenkov, D. V.: Quasivelocities and symmetries in non-holonomic systems. Dyn. Syst. 24(2), 187–222 (2009)

    MathSciNet  Article  MATH  Google Scholar 

  11. 11.

    Bonnard, B., Chyba, M. Singular Trajectories and Their Role in Control THeory: Mathématiques et Applications. Springer, Berlin (2003)

  12. 12.

    Bonnard, B., Kupka, I.: Théorie des singularités de l’application entrée/sortie et optimalité des trajectoires singulières dans le problème du temps minimal. Forum Mathematicum 5, 111–159 (1993)

    MathSciNet  Article  MATH  Google Scholar 

  13. 13.

    Bryson, A. E.: Applied Optimal Control. Hemisphere, New York (1975)

    Google Scholar 

  14. 14.

    Bullo, F., Lewis, A. D. Texts in applied mathematics: Geometric control of mechanical systems. Springer, New York (2004)

  15. 15.

    Chitour, Y., Jean, F., Trélat, E.: Singular trajectories of control-affine systems. SIAM J. Control. Optim. 47(2), 1078–1095 (2008). doi:10.1137/060663003

    MathSciNet  Article  MATH  Google Scholar 

  16. 16.

    Chyba, M., Haberkorn, T., Smith, R. N., Wilkens, G. R.: Controlling a Submerged Rigid Body: A Geometric Analysis, pp 375–385. Springer, Berlin (2007). doi:10.1007/978-3-540-73890-9_30

    Google Scholar 

  17. 17.

    Chyba, M., Leonard, N. E., Sontag, E. D.: Singular trajectories in multi-input time-optimal problems: Application to controlled mechanical systems. J. Dyn. Control. Syst. 9(1), 103–129 (2003). doi:10.1023/A:1022159318457

    MathSciNet  Article  MATH  Google Scholar 

  18. 18.

    Delgado-Tellez, M., Ibort, A.: On the Geometry and Topology of Singular Optimal Control Problems and Their Solutions Proceedings of the 4th International Conference on Dynamical Systems and Differential Equations, pp 223–233 (2002)

    Google Scholar 

  19. 19.

    Gabasov, R., Kirillova, F. M.: High order necessary conditions for optimality. SIAM J. Control 10(1), 127–168 (1972). doi:10.1137/0310012

    MathSciNet  Article  MATH  Google Scholar 

  20. 20.

    Gabasov, R., Kirillova, F. M.: Singular Optimal Controls. Nauka (1973)

  21. 21.

    Giaquinta, M., Hildebrandt, S.: Calculus of Variations I. Springer-Verlag, Berlin (1996)

    Google Scholar 

  22. 22.

    Goh, B. S.: Necessary conditions for singular extremals involving multiple control variables. SIAM J. Control 4(4), 716–731 (1966). doi:10.1137/0304052

    MathSciNet  Article  MATH  Google Scholar 

  23. 23.

    Gorokhovik, V. V.: High-order necessary optiMality conditions for control problems with terminal constraints. Opt. Control Appl. Meth. 4(2), 103–127 (1983)

    MathSciNet  Article  MATH  Google Scholar 

  24. 24.

    Greenwood, T. D.: Advanced Dynamics. Cambridge University Press, New York (2003)

    Google Scholar 

  25. 25.

    Jacobson, D. H.: A new necessary condition of optimality for singular control problems. SIAM J. Control 7(4), 578–595 (1969). doi:10.1137/0307042

    MathSciNet  Article  MATH  Google Scholar 

  26. 26.

    Jakubczyk, B., Krynski, W., Pelletier, F.: Characteristic vector fields of generic distributions of corank 2. Annales de l’Institut Henri Poincare Non Linear Analysis 26(1), 23–38 (2009). doi:10.1016/j.anihpc.2007.05.006

    MathSciNet  Article  MATH  Google Scholar 

  27. 27.

    Jóźwikowski, M., Respondek, W.: A contact covariant approach to optimal control with applications to sub-riemannian geometry. Math. Control Signals Syst. 28(3), 27 (2016). doi:10.1007/s00498-016-0176-3

    MathSciNet  Article  MATH  Google Scholar 

  28. 28.

    Kane, T. R., Levinson, D. A.: Dynamics: Theory and Applications. McGraw Hill, New York (1985)

    Google Scholar 

  29. 29.

    Kelley, H. J.: A second variation test for singular extremals. AIAA J. 2(8), 1380–1382 (1964). doi:10.2514/3.2562

    MathSciNet  Article  MATH  Google Scholar 

  30. 30.

    Kelley, H. J., Kopp, R. E., Moyer, H. G. Singular extremals. In: Leitman, G. (ed.) : Topics in Optimization, pp 63–101. Academic Press, New York (1967)

    Google Scholar 

  31. 31.

    Kopp, R. E., Moyer, H. G.: Necessary conditions for singular extremals. AIAA J. 3(8), 1439–1444 (1965). doi:10.2514/3.3165

    Article  MATH  Google Scholar 

  32. 32.

    Krener, A. J.: The high order maximal principle and its application to singular extremals. SIAM J. Control. Optim. 15(2), 256–293 (1977). doi:10.1137/0315019

    MathSciNet  Article  MATH  Google Scholar 

  33. 33.

    L’Afflitto, A., Haddad, W. M. A variational approach to the fuel optimal control. In: Agarwal, R. (ed.) : Recent Advances in Aircraft Technology, Chap. 10, pp 221–248. InTech, Croatia (2012)

    Google Scholar 

  34. 34.

    L’Afflitto, A., Haddad, W. M.: Necessary conditions for control effort minimization of euler-lagrange systems AIAA Guidance, Navigation, and Control Conference, pp 1–18 (2015)

    Google Scholar 

  35. 35.

    Lamnabhi-Lagarrigue, F.: Singular optimal control problems: On the order of a singular arc. Syst. Control Lett. 9(2), 173–182 (1987)

    MathSciNet  Article  MATH  Google Scholar 

  36. 36.

    Lamnabhi-Lagarrigue, F., Stefani, G.: Singular optimal control problems: On the necessary conditions of optimality. SIAM J. Control. Optim. 28(4), 823–840 (1990). doi:10.1137/0328047

    MathSciNet  Article  MATH  Google Scholar 

  37. 37.

    Liu, W.: Abnormal extremals and optimality in sub-riemannian manifolds IEEE Conference on Decision and Control. doi:10.1109/CDC.1994.411090, vol. 2, pp 1957–1963 (1994)

  38. 38.

    Maruskin, J.: Introduction to Dynamical Systems and Geometric Mechanics. Solar Crest, San Jose (2012)

    Google Scholar 

  39. 39.

    Maruskin, J. M., Bloch, A. M.: The Boltzmann-Hamel equations for the optimal control of mechanical systems with nonholonomic constraints. Int. J. Robust Nonlinear Control. doi:10.1002/rnc.1598 (2010)

  40. 40.

    McDanell, J. P., Powers, W. F.: Necessary conditions joining optimal singular and nonsingular subarcs. SIAM J. Control 9(2), 161–173 (1971). doi:10.1137/0309014

    MathSciNet  Article  MATH  Google Scholar 

  41. 41.

    Montgomery, R.: Abnormal minimizers. SIAM J. Control. Optim. 32(6), 1605–1620 (1994). doi: doi:10.1137/S0363012993244945

    MathSciNet  Article  MATH  Google Scholar 

  42. 42.

    Neimark, J. I., Fufaev, N. A.: Dynamics of Nonholonimic Systems. American Mathematical Society, New York (1972)

    Google Scholar 

  43. 43.

    Paul, R. P.: Robot manipulators: Mathematics, Programming, and Control : the Computer Control of Robot Manipulators. MIT Press, Boston (1981)

    Google Scholar 

  44. 44.

    Pontryagin, L. S., Boltyanskii, V. G., Gamkrelidze, R. V., Mishchenko, E. F.: The Mathematical Theory of Optimal Processes. Interscience Publishers, New York (1962)

    Google Scholar 

  45. 45.

    Rao, A. V., Benson, D. A., Darby, C., Patterson, M. A., Francolin, C., Sanders, I., Huntington, G. T.: Algorithm 902: GPOPS, A MATLAB software for solving multiple-phase optimal control problems using the Gauss pseudospectral method. ACM Trans. Math. Softw. 37, 1–39 (2010). doi:10.1145/1731022.1731032

    Article  MATH  Google Scholar 

  46. 46.

    Rao, A. V., Benson, D. A., Darby, C. L., Huntington, G. T.: User’s Manual for GPOPS Version 4.x: A MATLAB Software for Solving Multiple-Phase Optimal Control Problems Using hp–Adaptive Pseudospectral Methods. Tech. rep. (2011)

  47. 47.

    Robbins, H. M.: A generalized Legendre-Clebsch condition for the singular cases of optimal control. IBM J. Res. Dev. 11(4), 361–372 (1967). doi:10.1147/rd.114.0361

    Article  MATH  Google Scholar 

  48. 48.

    Ross, I. M., Fahroo, F.: User’s Manual for DIDO 2001: A MATLAB Application for Solving Optimal Control Problems. Tech. Rep. AAS-01-03, Monterey (2001)

  49. 49.

    Shuster, M. D.: Survey of attitude representations. J. Astronaut. Sci. 11, 439–517 (1993)

    MathSciNet  Google Scholar 

  50. 50.

    Sontag, E. D.: Remarks on the time-optimal control of a class of Hamiltonian systems IEEE Conference on Decision and Control, vol. 1, pp 217–221 (1989)

  51. 51.

    Sontag, E. D., Sussmann, H. J.: Time-optimal control of manipulators Proceedings of the IEEE International Conference on Robotics and Automation, vol. 3, pp 1692–1697 (1986)

  52. 52.

    Speyer, J. L.: On the fuel optimality of cruise. J. Aircr. 10(12), 763–765 (1973). doi:10.2514/3.60304

    Article  Google Scholar 

  53. 53.

    Spreyer, J. L., Jacobson, D.: Necessary and sufficient conditions for optimally for singular control problems; a transformation approach. J. Math. Anal. Appl. 33(1), 163–187 (1971). doi:10.1016/0022-247X(71)90190-9

    MathSciNet  Article  Google Scholar 

  54. 54.

    Zelikin, M. I., Lokutsievskiy, L. V., Hildebrand, R.: Geometry of neighborhoods of singular trajectories in problems with multidimensional control. Proc. Steklov Inst. Math. 277(1), 67–83 (2012). doi:10.1134/S0081543812040062

    MathSciNet  Article  MATH  Google Scholar 

Download references

Acknowledgments

This work was supported in part by NOAA/Office of Oceanic and Atmospheric Research under NOAA-University of Oklahoma Cooperative Agreement #NA16OAR4320115, the U.S. Department of Commerce, and the Air Force Office of Scientific Research under Grant FA9550-16-1-0100.

Author information

Affiliations

Authors

Corresponding author

Correspondence to Andrea L’Afflitto.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

L’Afflitto, A., Haddad, W.M. Abnormal Optimal Trajectory Planning of Multi-Body Systems in the Presence of Holonomic and Nonholonomic Constraints. J Intell Robot Syst 89, 51–67 (2018). https://doi.org/10.1007/s10846-017-0556-z

Download citation

Keywords

  • Optimal trajectory planning
  • Singular controls
  • Pontryagin’s principle
  • Normal and abnormal optimal control